What's the "propagator for real particles"?

Question

The propagators in quantum field theory are directly related to transition amplitudes of the form $\langle{t',\vec{x}}|{t,\vec{x}}\rangle $ where $|t,\vec{x}\rangle \equiv \phi(t,\vec{x}) |0\rangle$. In these notes and here it is argued that the propagators, e.g. the Feynman propagator $$\Delta_F (x-y) = \int \frac{d^4k}{(2\pi)^4} \frac{e^{-ik(x-y)}}{k^2 - m^2 + i\epsilon}$$ that we usually consider in quantum field theory is not "the propagator for real particles".

So what's the propagator for real particles? I know the difference between real and virtual particles. My problem is that the sources linked above emphasize that the Feynman propagator is only valid for virtual particles and I want to understand why.

Answer 1

The Feynman propagator is a Green's function. This means we find $\delta(x_\mu-y_\mu)$ if we plug it into the Klein-Gordon equation. This implies that the Feynman propagator describes (a sequence of) field configuration that only exists in the presence of interactions and thus describes "virtual particles".

In contrast, the "fundamental" propagator (which also known as the Wightman function) \begin{align}D(t',\vec{x}', t,\vec{x}) \equiv \langle{{t',\vec{x}'}|{t,\vec{x}}}\rangle &= \int \frac{ \mathrm{d }k^3 }{(2\pi)^3 2\omega_{k} } {\mathrm{e }}^{i\Big( \omega_k \cdot ( t ' - t) -\vec k \cdot (\vec x ' -\vec x) \Big)} \notag \\[-1ex] &= \int \frac{ \mathrm{d }k^3 }{(2\pi)^3 2\omega_{k} } {\mathrm{e }}^{i k^\mu (x_\mu' -x_\mu) } \end{align} is indeed a solution of the free Klein-Gordon equation (i.e. the kernel). Hence, it's a (sequence of) field configuration that is realizable for free fields and thus describes how "real particles" propagate.